Optimal System And Group Invariant Solutions Of The Geometric Invariant Flow

The study of geometric invariant flow has been widely used in image processing and phase transition,etc.In this thesis,we give a systematic study on the group invariant solutions of a centro-affine invariant flow as well as a hyperbolic-type affine invariant flow.This thesis consists of four chapters:In Chapter 1,the research background and related preliminaries are introduced.The main work of this thesis is outlined.In Chapter 2,the group invariant solutions of nonlinear partial differential equation which arises from the centro-affine invariant flow is studied.Firstly,the symmetric group of the equation is obtained by using the Lie group theory.Then based on the ideas of Olsiannikov and Olver,an optimal system and its reduction equations are obtained.Finally,we discuss the corresponding group invariant solutions.In Chapter 3,we study the nonlinear partial differential equation which is corresponded to the hyperbolic affine invariant flow.The symmetry group is determined and an optimal system of the equation is found.Furthermore,the corresponding reduced equations and some group invariant solutions are obtained.In Chapter 4,we summarize the thesis.